Bright 2020 brauer manin obstruction erdos straus

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corollary_1_3: For an odd prime p and any natural-number solution of 4/p = 1/u1 + 1/u2 + 1/u3, some ratio u_i/u_j with i different from j is a p-adic unit, and for every such ratio the Legendre symbol of -u_i/u_j modulo p is -1.

corollary_1_4: If n is an odd square, then 4/n = 1/u1 + 1/u2 + 1/u3 has no natural-number solution with n dividing u1 and n coprime to u2u3, and none with n coprime to u1 and n dividing both u2 and u3.

theorem_1_1: For every n at least 2 the Brauer–Manin obstruction does not rule out natural number solutions of 4/n = 1/u1 + 1/u2 + 1/u3.

theorem_1_2: For odd n and any solution of 4/n = 1/u1 + 1/u2 + 1/u3 in natural numbers, the product over primes p dividing n of the Hilbert symbols (-u1/u3, -u2/u3)_p equals -1.

theorem_1_5: For odd n and any integer solution of 4/n = 1/u1 + 1/u2 + 1/u3 that is not a natural-number solution, the product over primes p dividing n of the Hilbert symbols (-u1/u3, -u2/u3)_p equals 1.

theorem_1_6: For the affine surface U_n given by 4u1u2u3 = n(u1u2 + u1u3 + u2u3), the quotient Br U_n / Br Q is cyclic of order 2, generated by the quaternion algebra (-u1/u3, -u2/u3).

theorem_1_7: For a normal surface U over a field of characteristic 0 with rational singularities and a desingularisation f of U, the induced map from Br U to the Brauer group of the desingularisation is surjective.

theorem_1_8: As printed, for every natural number n the Brauer set of the product of the positive real points of U_n and its p-adic integral points is non-empty, and it is a proper subset of that product; the proof needs a prime dividing n.

theorem_1_9: For every natural number n the rational points of the Erdős–Straus surface U_n do not have dense image in its Brauer set of adelic points, so the Brauer–Manin obstruction does not explain every failure of strong approximation.

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Bright, Martin and Loughran, Daniel, Brauer-Manin obstruction for Erdős-Straus surfaces. Bull. Lond. Math. Soc. 52 (2020), no. 4, 746--761.

The authors apply the Brauer-Manin obstruction to the affine surface U_n given by 4u_1u_2u_3 = n(u_1u_2 + u_1u_3 + u_2u_3), the geometric model of the Erdős-Straus equation 4/n = 1/u_1 + 1/u_2 + 1/u_3. Theorem 1.6 computes Br U_n / Br Q as Z/2Z, generated by the quaternion algebra (-u_1/u_3, -u_2/u_3), and Theorem 1.1 shows that for every n >= 2 there is no Brauer-Manin obstruction to natural number solutions, so this obstruction cannot be used to disprove the conjecture; Theorem 1.8 (1.4) states the precise form, printed for all n in N, though its proof needs a prime dividing n. Nonetheless the Brauer set is a proper subset (Theorem 1.8), an obstruction to strong approximation made explicit at the primes dividing n: Theorem 1.2 shows that for odd n and any natural number solution, the product over p | n of the Hilbert symbols (-u_1/u_3, -u_2/u_3)_p equals -1, while Theorem 1.5 shows the product equals 1 for integer solutions that are not natural number solutions. Corollary 1.3 specializes to odd primes n = p: some ratio u_i/u_j with i != j is a p-adic unit, and every such ratio has Legendre symbol (-u_i/u_j | p) = -1, which unifies quadratic reciprocity conditions of Yamamoto for p ≡ 1 mod 4 (Appendix A). Corollary 1.4 recovers Elsholtz and Tao's Proposition 1.6, excluding two divisibility patterns when n is an odd square. Theorem 1.9 shows the rational points are not dense in the Brauer set, and Theorem 1.7 is a general surjectivity result for Brauer groups of surfaces with rational singularities.

Source: https://arxiv.org/abs/1908.02526.

The copy read for this card is arXiv:1908.02526v2 (12 May 2020, 17 pages); the published version, Bull. Lond. Math. Soc. 52 (2020), no. 4, 746--761, DOI 10.1112/blms.12374 (Crossref record fetched), was not compared. Read status: claims checked. The statements of Section 1 (pp. 1--5) were read clause by clause, pp. 1--4 on the page images, and the short proofs of Sections 3.4 to 3.9 (pp. 13--15) followed; Section 2 and the local lemmas of Sections 3.1 to 3.3 were not checked. Result pages, by printed label and page:

• Theorem 1.1 (p. 1)
• Theorem 1.2 (p. 2)
• Corollary 1.3 (p. 2)
• Corollary 1.4 (p. 2)
• Theorem 1.5 (p. 2)
• Theorem 1.6 (p. 3)
• Theorem 1.7 (p. 4)
• Theorem 1.8 (p. 4)
• Theorem 1.9 (p. 4)

The arXiv record names arXiv's non-exclusive distribution license (arXiv:1908.02526), every other right reserved. The Crossref record of the published version (DOI 10.1112/blms.12374, read 2026-10-07) names CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/) for the version of record.

Bears on. #242: the paper's equation allows equal u_i, so its results apply to every solution of Problem 242. Theorems 1.1 and 1.8 show the Brauer-Manin obstruction gives no route to a disproof; Theorem 1.2 and Corollaries 1.3 and 1.4 give necessary conditions on solutions for odd n, odd primes n and odd squares n. No result proves existence of a solution or addresses distinctness.

No file of this source is held: the arXiv version read carries no license that permits its redistribution, and the published version, whose CC BY 4.0 license would, was not obtained; the card cites the edition it names above.